Polytope of Type {4,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*432a
if this polytope has a name.
Group : SmallGroup(432,530)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 18, 108, 54
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {4,12,2} of size 864
   {4,12,4} of size 1728
Vertex Figure Of :
   {2,4,12} of size 864
   {4,4,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*216
   3-fold quotients : {4,4}*144
   6-fold quotients : {4,4}*72
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12}*864a
   3-fold covers : {4,12}*1296, {12,12}*1296c, {12,12}*1296e
   4-fold covers : {4,12}*1728b, {4,24}*1728b, {8,12}*1728b, {4,24}*1728d, {8,12}*1728c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      27 facets:
         27 of {4}*8
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s0*s1*s0*s1> of order 2.
      30 facets:
         6 of {2}*4
         24 of {4}*8
      10 vertex figures:
         8 of {12}*24
         2 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {4}*8
      6 vertex figures:
         6 of {12}*24
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      15 facets:
         3 of {2}*4
         12 of {4}*8
      5 vertex figures:
         4 of {12}*24
         1 of {6}*12
   P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2> of order 6.
      12 facets:
         6 of {2}*4
         6 of {4}*8
      4 vertex figures:
         2 of {12}*24
         2 of {6}*12

Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53);;
s1 := ( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(13,14)(16,24)(17,23)(18,22)(25,26)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(40,41)(43,51)(44,50)(45,49)(52,53);;
s2 := ( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(54)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53);
s1 := Sym(54)!( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(13,14)(16,24)(17,23)(18,22)(25,26)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(40,41)(43,51)(44,50)(45,49)(52,53);
s2 := Sym(54)!( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53);
poly := sub<Sym(54)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 >;

References : None.
to this polytope

Polytope of Type {4,12}

Twisty Puzzle